Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves

Abstract

We study the Harper-Hofstadter Hamiltonian and its corresponding non-perturbative butterfly spectrum. The problem is algebraically solvable whenever the magnetic flux is a rational multiple of 2π. For such values of the magnetic flux, the theory allows a formulation with two Bloch or θ-angles. We treat the problem by the path integral formulation, and show that the spectrum receives instanton corrections. Instantons as well as their one loop fluctuation determinants are found explicitly and the finding is matched with the numerical band width of the butterfly spectrum. We extend the analysis to all 2instanton sectors with different θ-angle dependence to leading order and show consistency with numerics. We further argue that the instanton-anti-instanton contributions are ambiguous and cancel the ambiguity of the perturbation series, as they should. We hint at the possibility of exact 2-instanton solutions responsible for such contributions via Picard-Lefschetz theory. We also present a powerful way to compute the perturbative fluctuations around the 1-instanton saddle as well as the instanton-anti-instanton ambiguity by using the topological string formulation.Emails:

Bloch electrons on honeycomb lattice and toric Calabi-Yau geometry

“…Using this nice connection, we can predict the higher order corrections to P inst top (n, φ) and P inst bot (n, φ) by using the topological string technique. The similar approach in the Hofstadter model is found in [8].…”

“…In this paper we proposed a new connection between the honeycomb lattice model and topological string theory. It is a non-trivial generalization of the original proposal in [2,8]. The non-perturbative corrections to the spectrum near the top or the bottom can be expressed by the NS free energy on local B 3 geometry.…”

“…From the results in the previous section, one easily finds A top (n, φ) and A bot (n, φ) as a series expansion in φ, It is claimed in [8] that these functions are related to the free energy of the refined topological string in the Nekrasov-Shatashvili limit at conifold singular points. This is natural because, as discussed in [2], the conifold singular points corresponds to the band edges.…”

“…We will briefly review the refined topological string in appendix A. One of the main results in [8] is the following relation between the NS free energy of local F 0 in the conifold frame 1 F c (t c , ) and the function A(n, φ) in the Hofstadter model JHEP05(2020)026 on square lattice,…”

Quantum periods and spectra in dimer models and Calabi-Yau geometries

“…The blowup equations originally come from studies of Seiberg-Witten gauge theories [24] (see also [25,26]), but have now become a very effective tool for computing topological string amplitudes on various Calabi-Yau manifolds [27][28][29][30][31]. The exact quantization conditions have also been applied to related condensed matter systems, e.g., in [32][33][34][35].…”